Is it correct to say that the union of infinite sets is a countable infinity?

Be careful here. If you have a set $\displaystyle \mathcal{N}=\left\{S_n:n\in\mathbb{N}\right\}$ then clearly $\displaystyle \mathcal{N}$ is countable for the "worst case scenario" is that all the sets are distinct in which case the mappking $\displaystyle f:\mathbb{N}\mapsto\mathcal{N}$ given by $\displaystyle f(n)=S_n$ is clearly a bijection. Saying that given a countable class of sets the union is automatically countable is just plain wrong. What about $\displaystyle \mathcal{N}=\left\{\mathbb{R}^n:n\in\mathbb{N}\rig ht\}$?